Foliation

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For other uses, see Foliation (disambiguation).

In mathematics, a foliation is a geometric device used to study manifolds. Informally speaking, a foliation is a kind of "clothing" worn on a manifold, cut from a striped fabric. On each sufficiently small piece of the manifold, these stripes give the manifold a local product structure. This product structure does not have to be consistent outside local patches (i.e., well-defined globally): a stripe followed around long enough might return to a different, nearby stripe.

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[edit] Definition

More formally, a dimension p foliation F of an n-dimensional manifold M is a covering by charts Ui together with maps

\phi_i:U_i \to \R^n

such that on the overlaps U_i \cap U_j the transition functions \varphi_{ij}:\mathbb{R}^n\to\mathbb{R}^n defined by

\varphi_{ij} =\phi_j \phi_i^{-1}

take the form

\varphi_{ij}(x,y) = (\varphi_{ij}^1(x),\varphi_{ij}^2(x,y))

where x denotes the first n − p co-ordinates, and y denotes the last p co-ordinates. That is,

\varphi_{ij}^1:\mathbb{R}^{n-p}\to\mathbb{R}^{n-p}

and

\varphi_{ij}^2:\mathbb{R}^p\to\mathbb{R}^{p}.

In the chart Ui, the stripes x = constant match up with the stripes on other charts Uj. Technically, these stripes are called plaques of the foliation. In each chart, the plaques are n − p dimensional submanifolds. These submanifolds piece together from chart to chart to form maximal connected injectively immersed submanifolds called the leaves of the foliation.


If we shrink the chart Ui it can be written in the form U_{ix}\times U_{iy} where U_{ix}\subset\mathbb{R}^{n-p} and U_{iy}\subset\mathbb{R}^p and Uiy is isomorphic to the plaques and the points of Uix parametrize the plaques in Ui. If we pick a y_0\in U_{iy}, U_{ix}\times\{y_0\} is a submanifold of Ui that intersects every plaque exactly once. This is called a local transversal section of the foliation. Note that due to monodromy there might not exist global transversal sections of the foliation.

[edit] Examples

[edit] Flat space

Consider an n-dimensional space, foliated as a product by subspaces consisting of points whose first n − p co-ordinates are constant. This can be covered with a single chart. The statement is essentially that

\mathbb{R}^n=\mathbb{R}^{n-p}\times \mathbb{R}^{p}

with the leaves or plaques \mathbb{R}^{n-p} being enumerated by \mathbb{R}^{p}. The analogy is seen directly in three dimensions, by taking n = 3 and p = 1: the two-dimensional leaves of a book are enumerated by a (one-dimensional) page number.

[edit] Covers

If M \to N is a covering between manifolds, and F is a foliation on N, then it pulls back to a foliation on M. More generally, if the map is merely a branched covering, where the branch locus is transverse to the foliation, then the foliation can be pulled back.

[edit] Submersions

If M^n \to N^q (where q \leq n ) is a submersion of manifolds, it follows from the inverse function theorem that the connected components of the fibers of the submersion define a codimension q foliation of M. Fiber bundles are an example of this type.

[edit] Lie groups

If G is a Lie group, and H is a subgroup obtained by exponentiating a closed subalgebra of the Lie algebra of G, then G is foliated by cosets of H.

[edit] Lie group actions

Let G be a Lie group acting smoothly on a manifold M. If the action is a locally free action or free action, then the orbits of G define a foliation of M.

[edit] Foliations and integrability

There is a close relationship, assuming everything is smooth, with vector fields: given a vector field X on M that is never zero, its integral curves will give a 1-dimensional foliation. (i.e. a codimension n − 1 foliation).

This observation generalises to a theorem of Ferdinand Georg Frobenius (the Frobenius theorem), saying that the necessary and sufficient conditions for a distribution (i.e. an n − p dimensional subbundle of the tangent bundle of a manifold) to be tangent to the leaves of a foliation, are that the set of vector fields tangent to the distribution are closed under Lie bracket. One can also phrase this differently, as a question of reduction of the structure group of the tangent bundle from GL(n) to a reducible subgroup.

The conditions in the Frobenius theorem appear as integrability conditions; and the assertion is that if those are fulfilled the reduction can take place because local transition functions with the required block structure exist.

There is a global foliation theory, because topological constraints exist. For example in the surface case, an everywhere non-zero vector field can exist on an orientable compact surface only for the torus. This is a consequence of the Poincaré-Hopf index theorem, which shows the Euler characteristic will have to be 0.

[edit] See also

[edit] References

  • Lawson, H. Blaine, "Foliations"
  • I.Moerdijk, J. Mrčun: Introduction to Foliations and Lie groupoids, Cambridge University Press 2003, ISBN 0521831970 (with proofs)
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